Character sums over unions of intervals

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Abstract

Let q be a cube-free positive integer and χ(modq) be a non-principal Dirichlet character. Our main result is a Burgess-type estimate for σ n∈A χ(n), where A ⊂[1,q] is the union of s disjoint intervals I 1, ⋯,I s. We obtain a nontrivial estimate for the character sum over A whenever |A|s -1/2 >q 1/4+ε and each interval Ij (1≤j≤s) has length |I j |>q ε for any ε>0. This follows from an improvement of a mean value Burgess-type estimate studied by Heath-Brown [Number Theory and Related Fields, Springer Proc. Math. Statist. 43, New York (2013), 199-213].

Original languageEnglish
Pages (from-to)3017-3026
Number of pages10
JournalForum Mathematicum
Volume27
Issue number5
DOIs
StatePublished - Sep 1 2015

Bibliographical note

Publisher Copyright:
© 2015 by De Gruyter.

Keywords

  • Character sum
  • harmonic analysis

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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